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Abstract

Optical frequency combs based on solitons in nonlinear microresonators
open up new regimes for optical metrology and signal processing across
a range of expanding and emerging applications. In this work, we
advance these combs toward applications by demonstrating protected
single-soliton formation and operation in a Kerr-nonlinear
microresonator using a phase-modulated pump laser. Phase modulation
gives rise to spatially/temporally varying effective loss and detuning
parameters, leading to an operation regime in which multi-soliton
degeneracy is lifted and a single soliton is the only observable
behavior. We achieve direct, on-demand excitation of single solitons
as indicated by reversal of the characteristic “soliton
step.” Phase modulation also enables precise, high bandwidth
control of the soliton pulse train’s properties, and we measure
dynamics that agree closely with simulations. We show that the
technique can be extended to high-repetition-frequency Kerr solitons
through subharmonic phase modulation. These results will facilitate
straightforward generation and control of Kerr-soliton microcombs for
integrated photonics systems.

Figures (6)

Fig. 1. (a) Schematic for soliton generation in a PM-pumped resonator,
neglecting interference at the output. (b) Simulated energy-level
diagrams for the CW- (orange) and PM-pumped (blue,
δPM=π) resonator for
F2=4, β2=−0.0187. With PM, an interval in
α exists for which the single soliton
is the only available energy level. This interval is fairly
narrow, but we find that it is readily accessible in experiment.
We also observe non-stationary states for values of
α≤2.7 in the PM case (red shading); whether
the system exhibits one of these non-stationary states or an
N=2 soliton state is determined by the
initial conditions that seed the formation of the soliton
ensemble.

Fig. 2. (a) Simulated quasi-CW background intensity without (orange) and
with PM of depth π/2 (blue) and 4π (green), with
F2=3 and β2=−0.02, with analytical approximations in
black. Here α is slightly larger than
α1, the critical value for soliton
formation. Dashed traces show the simulated phase profile of the
field ψ (green) and of the driving term
FeiδPMcosθ (red) with modulation depth
4π. The phase of the field is very
nearly the phase of the drive plus a constant offset. (b) A
simulation of spontaneous single-soliton generation using a
phase-modulated pump laser with δPM=π, F2=4, and β=−0.0187. Left: α is decreased smoothly as a function
of time τ and then held constant after a
soliton is generated. Middle: false-color plot of the intensity in
the cavity as a function of time, with the final intensity shown
above. Right: false-color plot of the comb spectrum on a
logarithmic scale as a function of time, assuming a repetition
rate of 22 GHz. Final spectrum is shown above. (c) A simulation of
the collapse of a soliton ensemble to N=1 with initial
N=5 for α=2.8, F2=4, and β=−0.0187. To slow down the initial dynamics
for clarity, the modulation depth is initialized at
δPM=0, and then linearly ramped to
δPM=π from τ=50 to τ=550. We note that the simulations
presented in parts (b) and (c) are similar to simulations
presented in Ref. [16]; we
present these for clarity without claiming novelty. (d) Plots of
approximations (obtained as described in the text,
β2=−0.0187) to the values
α1 and α2 of detuning at which the first
soliton and a second soliton are generated, respectively, with a
zoomed plot depicting details in the upper right. Several values
of PM depth are plotted: δPM=0 (blue), δPM=π (yellow), and
δPM=2π (green). For each the line to the
right is α1 and the line to the left is
α2; the single line for
δPM=0 represents the line at which the
bistability of the CW background vanishes everywhere in this case.
Horizontal lines indicate the interval α2<α<α1 obtained in the corresponding LLE
simulations with F2=5.1.

Fig. 3. (a) Frequency-domain depiction of the experiment, with cavity modes
shown in blue and laser frequencies shown in red. Modulation of a
counter-propagating probe beam at fPDH after shifting by
fAOM and subsequent locking of the red PDH
sideband to the resonance allows detuning control via
ν0−νpump=fAOM−fPDH, with adjustments implemented by
changing fPDH. (b) Schematic depiction of the
components of the experiment used for frequency control of the
system.

Fig. 4. (a) Measurement of a reversed soliton step in the comb power
associated with soliton generation from the background. Inset
shows qualitatively similar dynamics observed in an LLE-simulated
comb-power trace. (b) Measured optical spectrum of the soliton
generated with a phase-modulated pump laser. The spectrum of the
pump, which contains phase-modulation sidebands, is visible in the
center. The soliton’s spectral envelope closely matches the
sech2 envelope that has been overlaid in
dashed red. (c) Plot of the out-coupled soliton pulse
train’s repetition rate as recorded by a photodetector,
exhibiting a characteristically high signal-to-noise ratio. We
emphasize that this data is obtained with the phase modulation
turned off; otherwise the recorded RF signal is dominated by
through-coupled phase-modulation sidebands at
fPM∼fFSR∼frep. (d) Histogram of measured offset in
detuning from a reference value at which a soliton is generated
over 160 successive trials, with a Gaussian fit shown in red. The
width of the interval over which solitons are generated is larger
than the calculated width of the protected
N=1 level shown in Fig. 1(b), but the model does not
include laser fluctuations and other experimental effects.

Fig. 5. (a) Measured spectrogram of frep as fPM is swept over
±50kHz, with glitches where the locking
range is exceeded. Inset: qualitative agreement with simulations,
shown in red, when fPM is outside of the locking bandwidth.
As the soliton and the pump phase evolve at different frequencies
frep and fPM, the soliton periodically approaches
the maximum of the phase profile. The soliton’s group
velocity changes, nearly locking to the phase modulation, before
becoming clearly unlocked again. (b) Measured eye diagram of
frep as fPM is switched ±40kHz with 10 μs transition time.
(c) The same with 60 ns transition time, and an LLE simulation of
the dynamics (red) with depth δPM=0.9π and resonator linewidth
Δν=1.5MHz. (d) Simulated switching dynamics for
various linewidths and modulation depths. The theory trace from
(c) is reproduced in solid red in both panels. Top, solid black:
modulation depth 0.9π as in (c), and
Δν=10MHz (fastest, left), 3 MHz, and 1.4 MHz.
Bottom, dashed red: linewidth Δν=1.5MHz as in (c), with modulation depths of
2π and 6π (curves nearly overlap).

Fig. 6. (a) Simulated spectra of PM at fFSR with depth 0.15π (blue) and at
fFSR/21 with depth ∼8.3π (red). The relationships between the
fields that address resonator modes −1, 0, and 1 (as indicated by the gray
Lorentzian curves) are the same in both cases. (b) LLE simulation
of single-soliton generation using the subharmonic
phase-modulation spectrum shown in red in panel (a). Only modes
n=0,±21,±42,… of the phase-modulated driving field
are coupled into the resonator and affect the LLE dynamics, with
modes |n|>21 having negligible power. As
α is decreased from a large initial
value, a soliton is spontaneously generated, as in the case of
phase modulation near the FSR.